The Stacks project

Lemma 67.23.2. Let $S$ be a scheme. Let $X$ be a Jacobson algebraic space over $S$. For $x \in X_{\text{ft-pts}}$ and $g : W \to X$ locally of finite type with $W$ a scheme, if $x \in \mathop{\mathrm{Im}}(|g|)$, then there exists a closed point of $W$ mapping to $x$.

Proof. Let $U \to X$ be an ├ętale morphism with $U$ a scheme and with $u \in U$ closed mapping to $x$, see Morphisms of Spaces, Lemma 66.25.3. Observe that $W$, $W \times _ X U$, and $U$ are Jacobson schemes by Lemma 67.23.1. Hence finite type points on these schemes are the same thing as closed points by Morphisms, Lemma 29.16.8. The inverse image $T \subset W \times _ X U$ of $u$ is a nonempty (as $x$ in the image of $W \to X$) closed subset. By Morphisms, Lemma 29.16.7 there is a closed point $t$ of $W \times _ X U$ which maps to $u$. As $W \times _ X U \to W$ is locally of finite type the image of $t$ in $W$ is closed by Morphisms, Lemma 29.16.8. $\square$

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